| L(s) = 1 | − 2-s + 3-s − 5-s − 6-s − 7-s + 8-s + 10-s + 3·11-s + 4·13-s + 14-s − 15-s − 16-s + 2·17-s + 2·19-s − 21-s − 3·22-s − 23-s + 24-s + 5·25-s − 4·26-s − 27-s − 14·29-s + 30-s − 7·31-s + 3·33-s − 2·34-s + 35-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 0.447·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.316·10-s + 0.904·11-s + 1.10·13-s + 0.267·14-s − 0.258·15-s − 1/4·16-s + 0.485·17-s + 0.458·19-s − 0.218·21-s − 0.639·22-s − 0.208·23-s + 0.204·24-s + 25-s − 0.784·26-s − 0.192·27-s − 2.59·29-s + 0.182·30-s − 1.25·31-s + 0.522·33-s − 0.342·34-s + 0.169·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 933156 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 933156 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.257472933\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.257472933\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| 23 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 9 T + 28 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 9 T + 22 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 6 T - 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 10 T + 33 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 15 T + 146 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.20320780789765225110446586581, −9.629451573649923335150980265556, −9.285192270259309433152250942349, −8.990464854579590107304891052584, −8.712364128495663211896807800994, −8.151611972253176742407187286930, −7.891537584264714559627735374903, −7.22034971420824515605337767721, −7.16716295638896830820786596010, −6.40974687769886499293627151330, −6.12137083800776495148739351435, −5.39287542387732173532036117338, −5.12200821235566040316104309564, −4.24916017516082147188017778464, −3.84109608454822358073086456610, −3.27550290087565972836322940897, −3.22012272079427468943625547674, −1.85660060654480228580367842629, −1.67031298452011834084374658265, −0.58384491166812376422974592835,
0.58384491166812376422974592835, 1.67031298452011834084374658265, 1.85660060654480228580367842629, 3.22012272079427468943625547674, 3.27550290087565972836322940897, 3.84109608454822358073086456610, 4.24916017516082147188017778464, 5.12200821235566040316104309564, 5.39287542387732173532036117338, 6.12137083800776495148739351435, 6.40974687769886499293627151330, 7.16716295638896830820786596010, 7.22034971420824515605337767721, 7.891537584264714559627735374903, 8.151611972253176742407187286930, 8.712364128495663211896807800994, 8.990464854579590107304891052584, 9.285192270259309433152250942349, 9.629451573649923335150980265556, 10.20320780789765225110446586581
